The Theorem REF  is proved by using Crandall-Rabinowitz's Theorem REF  in the spirit of the previous works mentioned above. For that purpose, we reformulate the vortex patch equation (REF ) in the uniformly rotating framework with conformal mappings. We chose to use the conformal functional setting in order to be able to take advantage of the computations already done in [1]}, [2]}. Nevertheless, one could also use the polar parametrization similarly to the next result on quasi-periodic solutions. Introducing, for an initial domain \(D_0\)  close to the unit disc \(\mathbb {D}\) , the conformal parametrization \(\Phi :\mathbb {C}\setminus \overline{\mathbb {D}}\rightarrow \mathbb {C}\setminus \overline{D_0}\)  in the form
\(\Phi (z)=z+f(z),\qquad f(z)=\sum _{n=0}^{\infty }\frac{a_n}{z^n},\qquad a_n\in \mathbb {R},\) 
